Risk contagion under regular variation and asymptotic tail independence

Abstract

Risk contagion concerns any entity dealing with large scale risks. Suppose Z=(Z1,Z2) denotes a risk vector pertaining to two components in some system. A relevant measurement of risk contagion would be to quantify the amount of influence of high values of Z2 on Z1. This can be measured in a variety of ways. In this paper, we study two such measures: the quantity E{(Z1−t)+|Z2>t} called Marginal Mean Excess (MME) as well as the related quantity E(Z1|Z2>t) called Marginal Expected Shortfall (MES). Both quantities are indicators of risk contagion and useful in various applications ranging from finance, insurance and systemic risk to environmental and climate risk. We work under the assumptions of multivariate regular variation, hidden regular variation and asymptotic tail independence for the risk vector Z. Many broad and useful model classes satisfy these assumptions. We present several examples and derive the asymptotic behavior of both MME and MES as the threshold t→∞. We observe that although we assume asymptotic tail independence in the models, MME and MES converge to infinity under very general conditions; this reflects that the underlying weak dependence in the model still remains significant. Besides the consistency of the empirical estimators we introduce an extrapolation method based on extreme-value theory to estimate both MME and MES for high thresholds t where little data are available. We show that these estimators are consistent and illustrate our methodology in both simulated and real data sets.